If $a=cos~\alpha+i~sin~\alpha,$ $b=cos~\beta+i~sin~\beta$, $c=cos~\gamma+i~sin~\gamma$ and $\frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1$, then $cos(\beta-\gamma)+cos(\gamma-\alpha)+cos(\alpha-\beta)=$

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Accepted Answer

To solve the problem, we need to analyze the given expressions and conditions. We have:

$$ a = \cos \alpha + i \sin \alpha, $$
$$ b = \cos \beta + i \sin \beta, $$
$$ c = \cos \gamma + i \sin \gamma. $$

These are complex numbers on the unit circle, which can also be expressed in exponential form as:

$$ a = e^{i\alpha}, $$
$$ b = e^{i\beta}, $$
$$ c = e^{i\gamma}. $$

The condition given is:

$$ \frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1. $$

Substituting the exponential forms, we have:

$$ \frac{e^{i\beta}}{e^{i\gamma}} + \frac{e^{i\gamma}}{e^{i\alpha}} + \frac{e^{i\alpha}}{e^{i\beta}} = e^{i(\beta-\gamma)} + e^{i(\gamma-\alpha)} + e^{i(\alpha-\beta)} = 1. $$

This expression can be rewritten using Euler's formula:

$$ \cos(\beta-\gamma) + i\sin(\beta-\gamma) + \cos(\gamma-\alpha) + i\sin(\gamma-\alpha) + \cos(\alpha-\beta) + i\sin(\alpha-\beta) = 1. $$

For the real part of this equation, we have:

$$ \cos(\beta-\gamma) + \cos(\gamma-\alpha) + \cos(\alpha-\beta) = 1. $$

This matches the condition given in the problem. Therefore, the sum of the cosines of the differences of the angles is indeed 1.

Now, let's verify why Option 2 is correct and why the other options are incorrect:

1. **Option 1 (0):** This would imply that the sum of the cosines of the angle differences is zero, which contradicts the given condition that the sum is 1.

2. **Option 2 (1):** This is the correct answer, as shown by our derivation. The condition given in the problem directly leads to this result.

3. **Option 3 ($\frac{3}{2}$):** This would imply a larger sum than what is derived from the condition. The equation derived from the problem clearly shows the sum is 1, not $\frac{3}{2}$.

4. **Option 4 ($\frac{-2}{3}$):** This negative value is not possible given the context of the problem, as the sum of the cosines of angle differences derived from the unit circle condition is positive and equals 1.

Thus, the correct answer is indeed Option 2, as it satisfies the condition derived from the given equation.